Pronic Determinants of Circulant Matrices

Consider the number 28235 that is my diurnal age today. It has a circulant matrix with a determinant 10100 that is a pronic number since 10100 = 100 x 101.$$\begin{bmatrix}

2 & 8 & 2 & 3 & 5 \\

8 & 2 & 3 & 5 & 2 \\

2 & 3 & 5 & 2 & 8 \\

3 & 5 & 2 & 8 & 2 \\

5 & 2 & 8 & 2 & 3

\end{bmatrix}$$What's interesting is that most of the permutations of the digits of 28235 have determinants of their circulant matrices that are also pronic (permalink).

Number       | Determinant     | Factorisation  
------------------------------------------------
22358        | 15500           | 124 x 125      
22385        | 10100           | 100 x 101      
22538        | 19100           |                
22583        | 10100           | 100 x 101      
22835        | 19100           |                
22853        | 15500           | 124 x 125      
23258        | 19100           |                
23285        | 19100           |                
23528        | 10100           | 100 x 101      
23582        | 15500           | 124 x 125      
23825        | 15500           | 124 x 125      
23852        | 10100           | 100 x 101      
25238        | 15500           | 124 x 125      
25283        | 15500           | 124 x 125      
25328        | 10100           | 100 x 101      
25382        | 19100           |                
25823        | 19100           |                
25832        | 10100           | 100 x 101      
28235        | 10100           | 100 x 101      
28253        | 10100           | 100 x 101      
28325        | 15500           | 124 x 125      
28352        | 19100           |                
28523        | 19100           |                
28532        | 15500           | 124 x 125      
32258        | 10100           | 100 x 101      
32285        | 15500           | 124 x 125      
32528        | 15500           | 124 x 125      
32582        | 19100           |                
32825        | 10100           | 100 x 101      
32852        | 19100           |                
35228        | 19100           |                
35282        | 10100           | 100 x 101      
35822        | 15500           | 124 x 125      
38225        | 19100           |                
38252        | 15500           | 124 x 125      
38522        | 10100           | 100 x 101      
52238        | 10100           | 100 x 101      
52283        | 19100           |                
52328        | 19100           |                
52382        | 15500           | 124 x 125      
52823        | 10100           | 100 x 101      
52832        | 15500           | 124 x 125      
53228        | 15500           | 124 x 125      
53282        | 10100           | 100 x 101      
53822        | 19100           |                
58223        | 15500           | 124 x 125      
58232        | 19100           |                
58322        | 10100           | 100 x 101      
82235        | 15500           | 124 x 125      
82253        | 19100           |                
82325        | 19100           |                
82352        | 10100           | 100 x 101      
82523        | 15500           | 124 x 125      
82532        | 10100           | 100 x 101      
83225        | 10100           | 100 x 101      
83252        | 15500           | 124 x 125      
83522        | 19100           |                
85223        | 10100           | 100 x 101      
85232        | 19100           |                
85322        | 15500           | 124 x 125  

Note that it is only when the determinant is 19100 that it is not pronic since 19100 = 100 x 191. In my post titled Determinants of Circulant Matrices, I listed all numbers up to 40000 with the property that the determinants of their circulant matrices were pronic. The numbers between 28000 and 40000 are:

28235, 28253, 28325, 28327, 28453, 28479, 28532, 28543, 28574, 28619, 28732, 28776, 29054, 29168, 29245, 29254, 29555, 29700, 29748, 30171, 30179, 30566, 30575, 30665, 30900, 31100, 31107, 31134, 31233, 31323, 31332, 31355, 31358, 31385, 31400, 31413, 31422, 31440, 31510, 31637, 31646, 31684, 31763, 31907, 32124, 32133, 32223, 32232, 32241, 32258, 32285, 32287, 32313, 32322, 32331, 32528, 32728, 32825, 32845, 32854, 32960, 33123, 33132, 33141, 33176, 33213, 33222, 33231, 33312, 33321, 33335, 33353, 33515, 33518, 33533, 33569, 33671, 33789, 33815, 33965, 33987, 34100, 34166, 34212, 34258, 34311, 34410, 34582, 34599, 34700, 34861, 35153, 35183, 35248, 35282, 35333, 35482, 35507, 35531, 35606, 35693, 35822, 35831, 35936, 35949, 35996, 35999, 36056, 36065, 36137, 36359, 36395, 36418, 36461, 36506, 36614, 36713, 36920, 36995, 37011, 37055, 37091, 37316, 37361, 37400, 37700, 37799, 37822, 37893, 37938, 37979, 38146, 38153, 38252, 38272, 38379, 38397, 38425, 38522, 38524, 38531, 39495, 39536, 39569, 39599, 39653, 39659, 39738, 39797, 39873, 39900, 39954, 39959, 39977, 39995

Note that with 28235 and its digit permutations there are TWO determinants that satisfy. These are:

• \(10100 = 100 \times 101\)
• \(15500 = 124 \times 125\)

This is generally not the case. Consider 28327 and its digit permutations where only the determinant 14762 = 121 x 122 satisfies (permalink).

Number       | Determinant     | Factorisation  
------------------------------------------------
22378        | 14762           | 121 x 122      
22387        | 11462           |                
22738        | 27962           |                
22783        | 11462           |                
22837        | 27962           |                
22873        | 14762           | 121 x 122      
23278        | 27962           |                
23287        | 27962           |                
23728        | 11462           |                
23782        | 14762           | 121 x 122      
23827        | 14762           | 121 x 122      
23872        | 11462           |                
27238        | 14762           | 121 x 122      
27283        | 14762           | 121 x 122      
27328        | 11462           |                
27382        | 27962           |                
27823        | 27962           |                
27832        | 11462           |                
28237        | 11462           |                
28273        | 11462           |                
28327        | 14762           | 121 x 122      
28372        | 27962           |                
28723        | 27962           |                
28732        | 14762           | 121 x 122      
32278        | 11462           |                
32287        | 14762           | 121 x 122      
32728        | 14762           | 121 x 122      
32782        | 27962           |                
32827        | 11462           |                
32872        | 27962           |                
37228        | 27962           |                
37282        | 11462           |                
37822        | 14762           | 121 x 122      
38227        | 27962           |                
38272        | 14762           | 121 x 122      
38722        | 11462           |                
72238        | 11462           |                
72283        | 27962           |                
72328        | 27962           |                
72382        | 14762           | 121 x 122      
72823        | 11462           |                
72832        | 14762           | 121 x 122      
73228        | 14762           | 121 x 122      
73282        | 11462           |                
73822        | 27962           |                
78223        | 14762           | 121 x 122      
78232        | 27962           |                
78322        | 11462           |                
82237        | 14762           | 121 x 122      
82273        | 27962           |                
82327        | 27962           |                
82372        | 11462           |                
82723        | 14762           | 121 x 122      
82732        | 11462           |                
83227        | 11462           |                
83272        | 14762           | 121 x 122      
83722        | 27962           |                
87223        | 11462           |                
87232        | 27962           |                
87322        | 14762           | 121 x 122   

Recurring Digital Invariant Variant (RDIV) Algorithm

Let's consider the following algorithm (formally called the Recurring Digital Invariant Variant or RDIV algorithm - see this link for an explanation of the name):

• choose a number \(n\)
• let \(k\) be the number of digits in \(n\)
• raise each digit of \(n\) to the \(k\)-th power and add the results
• call the new number \(n\) and repeat

Let's use \(n=14\) as an example:

• \(14 \rightarrow 1^2 + 4^2 = 17\)
• \(17 \rightarrow 1^2 + 7^2 = 50\)
• \(50 \rightarrow 5^2 + 0^2 = 25\)
• \(25 \rightarrow 2^2 + 5^2 = 29\)
• \(29 \rightarrow 2^2 + 9^2 = 85\)
• \(85 \rightarrow 8^2 + 5^2 = 89\)
• \(89 \rightarrow 8^2 + 9^2 = 145\)
• \(145 \rightarrow 1^3 + 4^3 + 5^3 = 190\)
• \(190 \rightarrow 1^3 + 9^3 + 0^3 = 730\)
• \(730 \rightarrow 7^3 + 3^3 + 0^3 = 370\)
• \(370 \rightarrow 3^3 + 7^3 + 0^3 = 370\) 

370 is a narcissistic number as explained in my post Narcissistic, D-Powerfull and Friedman Numbers. The trajectory of any number under this algorithm will either end with a narcissistic number (as was the case with 14) or it will enter a loop (as is the case with 28218). The latter has the following trajectory (permalink):

==================================================

RDIV TRAJECTORY ANALYSIS FOR INPUT: 28218

==================================================

Full Trajectory Visited:

28218, 65601, 18678, 90120, 59082, 94974, 136953, 595181, 824837, 646826, 406272, 168529, 855931, 825565, 355739, 681798, 1220035, 80569, 102718, 379859, 1459029, 9660576, 6524445, 485466, 379273, 768261, 473170, 240124, 8321, 4194, 7074, 5058, 5346, 2258, 4753, 3363, 1539, 7268, 7809, 13058, 36137, 25070, 19964, 126899, 1371747, 2489202, 6896889, 16417266, 10869443, 61641187, 25966788, 86116067, 27580867, 47154531, 6683686, 5316235, 440689, 848433, 533938, 811397, 911965, 1125165, 436317, 169860, 886898, 1626673, 1665667, 2021413, 18829, 124618, 312962, 578955, 958109, 1340652, 376761, 329340, 537059, 681069 -> [loops back to 886898]

Loop Entry Point:   886898 (encountered at step 65)

Pre-period Length:  64 step(s) before entering cycle

Cycle Length:       14 distinct number(s) in the loop

Canonical Cycle:    18829, 124618, 312962, 578955, 958109, 1340652, 376761, 329340, 537059, 681069, 886898, 1626673, 1665667, 2021413

==================================================

Figure 1 shows a graph of its trajectory:

Figure 1: permalink

I've incorporated this algorithm into my daily number analysis.

Number Base Permutations

There are some numbers that have the same digits but in a different order when converted to another number base. Take 10144 as an example:$$10144_{10}= 41401_7$$The five digit numbers from 10000 to 40000 with this property are as follows (permalink):

10144, 10235, 10342, 10453, 10542, 11425, 11750, 12415, 12450, 12564, 12651, 13045, 13245, 13260, 13402, 13534, 13620, 14610, 15226, 15643, 15680, 16121, 16255, 16273, 16546, 16633, 18291, 19463, 19730, 21322, 21753, 21763, 21835, 23146, 23568, 24871, 25061, 25169, 26804, 26931, 29103, 29610, 30189, 30576, 31112, 31457, 31481, 32321, 32348, 34179, 34582, 35001, 35024, 35081, 35228, 35731, 36417, 37465, 38276

The table below shows the full details for the above numbers. As can be seen, they are not that frequent in the given range.

Results for range: 10000 to 40000
Decimal Number     | Base b Representation  | Base
-------------------------------------------------------
10144              | 41401                  | 7
10235              | 15032                  | 9
10342              | 42103                  | 7
10453              | 15304                  | 9
10542              | 42510                  | 7
11425              | 45211                  | 7
11750              | 17105                  | 9
12415              | 51124                  | 7
12450              | 51204                  | 7
12564              | 51426                  | 7
12651              | 51612                  | 7
13045              | 53014                  | 7
13245              | 53421                  | 7
13260              | 20163                  | 9
13402              | 20341                  | 9
13534              | 54313                  | 7
13620              | 20613                  | 9
14610              | 60411                  | 7
15226              | 62251                  | 7
15643              | 63415                  | 7
15680              | 10865                  | 11
16121              | 11126                  | 11
16255              | 65251                  | 7
16273              | 37621                  | 8
16546              | 66145                  | 7
16633              | 66331                  | 7
18291              | 12819                  | 11
19463              | 13694                  | 11
19730              | 13907                  | 11
21322              | 32221                  | 9
21753              | 52371                  | 8
21763              | 32761                  | 9
21835              | 32851                  | 9
23146              | 16432                  | 11
23568              | 35286                  | 9
24871              | 12487                  | 12
25061              | 12605                  | 12
25169              | 12695                  | 12
26804              | 40682                  | 9
26931              | 19263                  | 11
29103              | 10329                  | 13
29610              | 10629                  | 13
30189              | 10983                  | 13
30576              | 73560                  | 8
31112              | 11213                  | 13
31457              | 75341                  | 8
31481              | 11438                  | 13
32321              | 22313                  | 11
32348              | 48332                  | 9
34179              | 17943                  | 12
34582              | 52384                  | 9
35001              | 53010                  | 9
35024              | 24350                  | 11
35081              | 53108                  | 9
35228              | 53282                  | 9
35731              | 13357                  | 13
36417              | 13764                  | 13
37465              | 56347                  | 9
38276              | 26837                  | 11

Note that the bases range from 7 to 13. While this sequence does not occur in the OEIS, some subsets do. For example, OEIS  A037440:

 A037440: positive numbers having the same set of digits in bases 7 and 10.

Self-Fibonacci

Here is an interesting sequence generated by a hidden connection to Fibonacci based on the letter-number association shown in Table 1:

Table 1: source

The sequence is OEIS A129938:

A129938: "Self-Fibonacci"; a(n) is the sum of the last nine terms. Sequence starts with 6, 9, 2, 15, 14, 1, 3, 3, 9 which are f, i, b, o, n, a, c, c, i if you consider a=1, b=2, c=3, ..., z=26.

The sequence begins 6, 9, 2, 15, 14, 1, 3, 3, 9, 62, 118, 227, 452, 889, 1764, 3527, 7051, 14099, 28189, ...

I only chanced upon this sequence because 28189, my diurnal age today, is a member. I've written about the various connections between numbers and letters in a post titled Days of the Year and Gematria back in August of 2021. The idea behind this sequence reminds me of my own approach described in a blog post titled Consolidating Fibonacci-like Numbers where I considered numbers whose digits following a Fibonacci pattern e.g. 21347:$$21347 \text{ where }2 + 1 =3, 1+3=4,3+4=7$$However, getting back to approach followed in OEIS A129938, an interesting "spin-off" could be that previously unnamed tribonacci sequences could be given memorable names. For example, using Table 1 we could write:$$ \text{ cat } \rightarrow \text{ c, a, t } \rightarrow \text{ 3, 1, 20 }$$and so the "cat" sequence becomes:$$3, 1, 20, 24, 45, 79, \dots$$Similarly we have:$$ \text{ dog } \rightarrow \text{ d, o, g } \rightarrow \text{ 4, 15, 7 }$$ So the "dog" sequence becomes:$$4, 15, 7, 26, 48, 81, \dots$$Silly I know but it would make for an interesting puzzle in Puzzle of the Day. The sequence doesn't have to be tribonacci, it could simply be Fibonacci-like. For example, we could ask why is the sequence 13, 5, 18, 23, 41, 65, ... egocentric? The answer is that:$$13, 5 \rightarrow \text{ m, e } \rightarrow \text{ me }$$Similarly, the sequence could be made of four or more seeds and a puzzle created. For example, we could ask what does this sequence 13, 9, 12, 11, 45, 77, ... and the Milky Way have in common? The answer is that the first four members of the sequence are the seeds to generate the future members of the sequence and we have:$$ 13, 9, 12, 11 \rightarrow \text{ m, i, l, k } \rightarrow \text{ milk}$$That's enough nonsense for the moment but let's not forget that there has always been a long-standing connection between letters of certain alphabets (Hebrew, Ancient Greek and Arabic for example) and numbers. With the English language the connection has weakened but it's still there and not just in the way shown in Table 1. There are other ways to assign values to letters in the English alphabet. Table 2 shows an alternative way that is more in keeping with the ancient languages.

Table 2: source

Beatty's Theorem

This video on Beatty's Theorem by Euclidia is very interesting and very surprising. It shows that the integers can be split into two disjoint sets using an irrational number of your choice provided it is greater than 1. Each pair is unique to the irrational number being used. The two infinite sets cover the entire range of positive integers.

It's extremely easy to generate the two sequences using the following SageMath code. Here, using \( \sqrt{2} \), I've generated the first 25 terms of each sequence:

a=sqrt(2)
b=a/(a-1)
A,B=[],[]
for n in [1..25]:
    A.append(floor(n*a))
    B.append(floor(n*b))
print(A)

print(B)

[1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, 25, 26, 28, 29, 31, 32, 33, 35]
[3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, 47, 51, 54, 58, 61, 64, 68, 71, 75, 78, 81, 85] 

Using different values of \(a\) such as \( \pi \) or \(e\) yields different disjoint sets:

\( \pi\) yields the following sequences:

[3, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 47, 50, 53, 56, 59, 62, 65, 69, 72, 75, 78]

[1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36]

\(e\) yields the following sequences:

[2, 5, 8, 10, 13, 16, 19, 21, 24, 27, 29, 32, 35, 38, 40, 43, 46, 48, 51, 54, 57, 59, 62, 65, 67]

[1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, 30, 31, 33, 34, 36, 37, 39]

INTERACTIVE LINK

Here a link to an interactive report that Gemini created regarding Beatty's sequences. I struggled to get this to work in Blogger but by using this external link, it all worked fine. Something to remember in the future.

Numbers as Dates

When I turned 28000 days old, there was no way to turn this number into a date until I reached 28100 days old. At this point, it's possible via this stratagem:

28 - 1 - 00 --> 28th January 1900

There is ambiguity here because 00 could be interpreted as representing 2000 but I'll stick with 1900 for reasons that I'll make apparent. Yesterday I turned 28149 days old which converts as follows:

28 - 1 - 49 --> 28th January 1949

Hmmm. On that date, I was not yet born although I was a seven month old foetus. However, today I'm 28150 days old and this converts as follows:

28 - 1 - 50 --> 28th January 1950

Now by that date I had been born and was about ten months old. The interesting thing is that when I turn 30449 days old, this can be made to correspond to my birthday in the following way:

3 - 04 - 49 --> 3rd April 1949

There's ambiguity here because 30449 can also be made into another date:

30 - 4 - 49 --> 30th April 1949

All this requires that we admit to leading zeros when we want them, omit them when we don't and make choices to suit ourselves when confronted with competing alternatives. In other words, flexibility is required. This process is not continuous. It works until I exceed 28999 since:

28 - 9 - 99 --> 28th September 1999

However, 29000 doesn't work and this number to date conversion only becomes possible again when I reach 29100 since:

29 - 1 - 00 --> 29th January 1900

With the leading two digits being 29, care must be taken with leap years and February. What happens with 29200?

29 - 2 - 00 --> 29th February 1900

Is the 29th of February 1900 a valid date? It is if 1900 were a leap year but it's not since it's not divisible by 400. If we interpret the date as 29th February 2000 however, then the date is valid since 2000 was a leap year. Sticking with the twentieth century convention, valid dates only arise with 29204, 29208, 29212, ... , 29288, 29292, 29296.

After reaching 29300, there's no problem with the leading two digits being 29 and things will be fine until I reach 30000. The number to date conversion will kick in again at 30100 and I'll reach 30449 without any gaps. All this is hardly serious mathematics but fun nonetheless. 

In all of this, I've adopted a day - month - year format switching between DD-M-YY and D-MM-YY as it suits me. This works well whereas other formats lead to lots of inadmissable dates and aren't as much fun.

Digit Manipulation

Video created by NotebookLM based on blog content

DIGITS TO SQUARES

Yesterday I turned 28113 days old and this number is a member of OEIS A048383: numbers \(k\) such that replacing each nonzero digit \(d\) with the \(d\)-th prime (replacing each 0 digit with a 1) yields a square. So this means that:$$28113 \rightarrow 319225 = 5^2 \times 113^2$$The members of this sequence are few and far between and in the range up to 40000 only the following numbers satisfy (permalink):$$ \begin{align} 0 &\rightarrow 1 = 1\\13 &\rightarrow 25 = 5^2\\113 &\rightarrow 225 = 3^2 \times 5^2\\2410 &\rightarrow 3721 = 61^2\\4113 &\rightarrow 7225 = 5^2 \times 17^2\\6113 &\rightarrow 13225 = 5^2 \times 23^2\\8210 &\rightarrow 19321 = 139^2\\14113 &\rightarrow 27225 = 3^2 \times 5^2 \times 11^2\\23410 &\rightarrow 35721 = 3^6 \times 7^2\\28113 &\rightarrow 319225 = 5^2 \times 113^2\\33113 &\rightarrow 55225 = 5^2 \times 47^2\\34010 &\rightarrow 57121 = 239^2\\35113 &\rightarrow 511225 = 5^2 \times 11^2 \times 13^2\\\end{align}$$DIGITS TO PRIMES

A variation on this theme is OEIS A048381:

A048381: numbers \(k\) such that replacing each nonzero digit \(d\) with the \(d\)-th prime (replacing each 0 digit with a 1) yields a prime.

Members of this sequence are far more numerous with 5629 in the range up to 40000. An example is 28112 since:$$28112 \rightarrow 319223 \text{ which is prime} $$Some upcoming members are:

28124, 28146, 28152, 28155, 28202, 28210, 28214, 28216, 28226, 28228, 28230, 28234, 28235, 28236, 28247, 28265, 28270, 28277, 28289, 28294, 28295, 28298, 28300, 28317, 28319, 28328, 28329, 28344, 28359, 28360, 28368, 28388, 28392, 28397, 28414, 28418, 28422, 28429, 28434, 28449, 28458, 28464, 28470, 28474, 28485, 28490, 28498, 28502, 28504, 28515, 28524, 28525, 28529, 28546, 28562, 28575, 28592, 28599, 28606, 28612, 28614, 28622, 28630, 28652, 28658, 28665, 28667, 28674, 28684, 28686, 28706, 28717, 28724, 28744, 28752, 28772, 28786, 28807, 28810, 28814, 28825, 28827, 28838, 28854, 28868, 28870, 28876, 28886, 28888, 28890, 28928, 28929, 28932, 28948, 28955, 28960, 28966, 28979, 28984, 28988, 28995, 28997

One way to thin the numbers when there are so many in a given range is to require that the numbers come in pairs that are consecutive integers. If this requirement is imposed then the 5629 reduces to 580. Imposing the restriction that the numbers are triplets that are consecutive integers reduces the 580 further to a manageable 103:

1, 2, 3, 4, 5, 6, 7, 24, 25, 144, 166, 167, 414, 474, 506, 674, 897, 898, 1026, 1027, 1176, 1177, 1398, 1516, 1824, 2035, 2074, 2094, 2146, 2544, 3316, 4044, 5247, 5248, 5286, 5514, 6044, 6484, 7116, 7117, 7118, 7264, 7918, 8008, 8127, 8444, 8665, 10016, 11046, 11047, 11404, 13068, 13445, 14224, 14584, 15886, 16055, 16346, 16347, 16505, 16945, 18306, 18497, 19276, 19465, 20044, 20124, 21797, 21798, 22167, 22416, 22417, 22586, 22694, 22767, 23336, 23774, 24726, 24727, 24845, 25934, 26608, 26844, 26885, 28234, 29376, 29377, 29714, 29715, 29917, 30145, 30705, 32244, 32248, 33807, 35405, 35647, 36018, 36635, 37888, 38097, 39067, 39527

Let's take 28234 as an example where:$$ \begin{align} 28234 &\rightarrow 319357\\28235 &\rightarrow 3193511\\28236 &\rightarrow 3193513 \end{align}$$There is of course an initial run of seven numbers (1 to 7) and after that there are runs of four numbers beginning with:

24, 166, 897, 1026, 1176, 5247, 7116, 7117, 11046, 16346, 21797, 22416, 24726, 29376, 29714

Finally there is only one run of five numbers and it starts with 7116.

DIGITS TO PALINDROMES

Another variation, using this same method of digit manipulation, is to ask what non-palindromic number become palindromes. Well, in the range up to 40000, it turns out that 333 numbers satisfy this condition (permalink). The numbers from 28113 onwards are:

28086, 28586, 28686, 28786, 28802, 28886, 29029, 29069, 29129, 29199, 29212, 29229, 29329, 29429, 29529, 29569, 29612, 29669, 29769, 29869, 29912, 29999, 30053, 30553, 30653, 30753, 30853, 32063, 32193, 32563, 32663, 32763, 32863, 32993, 34073, 34573, 34673, 34773, 34873, 35003, 35503, 36203, 36603, 36903, 37403, 37703, 38803, 39213, 39613, 39913

Let's use 28086 as an example:$$28086 \rightarrow 31911913$$Clearly there are many possible variations using just this particular type of digit manipulation and in this post I've shown examples of three of them where the digits \(d\) are manipulated as follows:$$ \begin{align} d &\rightarrow \text{ prime}(d) \text{ if } d \neq 0 \\0 &\rightarrow 1 \end{align} $$However, other manipulations are limited only by your imagination. An example of different type of manipulation would be:$$ d \rightarrow d^{ \, \small{2}}$$We can ask how many non-palindromic numbers become palindromes when their digits are manipulated in this manner (squared). In the range up to 40000, the answer is that there are 96 suitable numbers and they are (permalink):

19, 28, 37, 41, 72, 199, 288, 327, 377, 441, 461, 732, 772, 1191, 1281, 1371, 1411, 1721, 1919, 1999, 2192, 2282, 2372, 2412, 2722, 2828, 2888, 3193, 3207, 3217, 3227, 3237, 3283, 3373, 3413, 3723, 3737, 3777, 4141, 4441, 4661, 7032, 7132, 7232, 7272, 7332, 7772, 11991, 12881, 13271, 13771, 14411, 14611, 17321, 17721, 19019, 19119, 19219, 19319, 19999, 21992, 22882, 23272, 23772, 24412, 24612, 27322, 27722, 28028, 28128, 28228, 28328, 28888, 31993, 32007, 32117, 32197, 32227, 32287, 32337, 32377, 32417, 32727, 32883, 33273, 33773, 34413, 34613, 37037, 37137, 37237, 37277, 37323, 37337, 37723, 37777

An example is 28028 where$$28028 \rightarrow 4640464$$

Why Is 29 Special?

 This little puzzle appeared in Puzzle A Day.

What is the only number that, when written in typed letters as all capitals, is made up of exactly the same number of straight-line segments as the value of the number itself? For example, FIVE uses 10 straight lines, so it cannot be the answer. (Ignore any hyphens in the spelled-out number.)

I asked Gemini to create a table for the number of straight line segments in each letter of the English alphabet and then solve the puzzle.

Python

def number_to_words(n):
    # Basic implementation for 0-99
    # Not handling large numbers as we expect small solutions
    ones = ["ZERO", "ONE", "TWO", "THREE", "FOUR", "FIVE", "SIX", "SEVEN", "EIGHT", "NINE"]
    teens = ["TEN", "ELEVEN", "TWELVE", "THIRTEEN", "FOURTEEN", "FIFTEEN", "SIXTEEN", "SEVENTEEN", "EIGHTEEN", "NINETEEN"]
    tens = ["", "", "TWENTY", "THIRTY", "FORTY", "FIFTY", "SIXTY", "SEVENTY", "EIGHTY", "NINETY"]
    
    if 0 <= n < 10:
        return ones[n]
    elif 10 <= n < 20:
        return teens[n-10]
    elif 20 <= n < 100:
        if n % 10 == 0:
            return tens[n // 10]
        else:
            return tens[n // 10] + " " + ones[n % 10]
    return "NUMBER_TOO_LARGE"

# Letter segment counts
# Assumption: Sans-serif, standard straight lines.
# B: 1 vertical
# D: 1 vertical
# G: 1 horizontal
# J: 0 (hook) or 1 (vertical)? Let's assume 0 first as usually curve.
# P: 1 vertical
# Q: 1 tail
# R: 2 (vertical + diagonal)
# U: 0 (curve)
# Y: 3
segments = {
    'A': 3, 'B': 1, 'C': 0, 'D': 1, 'E': 4, 'F': 3, 'G': 1, 'H': 3, 'I': 1, 
    'J': 0, 'K': 3, 'L': 2, 'M': 4, 'N': 3, 'O': 0, 'P': 1, 'Q': 1, 'R': 2, 
    'S': 0, 'T': 2, 'U': 0, 'V': 2, 'W': 4, 'X': 2, 'Y': 3, 'Z': 3
}

# Check matches
matches = []
for i in range(100):
    word = number_to_words(i)
    # Remove spaces and hyphens if any (my function adds space)
    clean_word = word.replace(" ", "").replace("-", "")
    count = 0
    for char in clean_word:
        if char in segments:
            count += segments[char]
    
    if count == i:
        matches.append((i, word, count))

print("Matches:", matches)

Code output

Matches: [(16, 'SIXTEEN', 16), (29, 'TWENTY NINE', 29)]

Based on a standard analysis of the straight line segments in the uppercase English alphabet (using a typical sans-serif block font), here is the breakdown.

Table of Straight Line Segments per Letter

Letter	Segments	Shape Breakdown
A	3	Two diagonal, one horizontal
B	1	One vertical (curves do not count)
C	0	All curved
D	1	One vertical
E	4	One vertical, three horizontal
F	3	One vertical, two horizontal
G	1	One short horizontal shelf
H	3	Two vertical, one horizontal
I	1	One vertical
J	0	Typically a hook (though some fonts use 1 vertical)
K	3	One vertical, two diagonal
L	2	One vertical, one horizontal
M	4	Two vertical, two diagonal
N	3	Two vertical, one diagonal
O	0	All curved
P	1	One vertical
Q	1	One short diagonal tail
R	2	One vertical, one diagonal leg
S	0	All curved
T	2	One vertical, one horizontal
U	0	Typically a curve (block versions may have 2 vertical)
V	2	Two diagonal
W	4	Four diagonal
X	2	Two diagonal
Y	3	Two diagonal, one vertical (or three meeting at center)
Z	3	Two horizontal, one diagonal

The Numbers That Match

Using the values from the table above, we can deduce which integers (when written as English words) have a value equal to their total number of straight line segments.

There are exactly two integers that satisfy this condition:

1. SIXTEEN

• S (0) + I (1) + X (2) + T (2) + E (4) + E (4) + N (3)

• Calculation: $0 + 1 + 2 + 2 + 4 + 4 + 3 = \mathbf{16}$

• Result: The word "SIXTEEN" is made of 16 straight lines, which equals its value.

2. TWENTY NINE

• TWENTY: T (2) + W (4) + E (4) + N (3) + T (2) + Y (3) = 18

• NINE: N (3) + I (1) + N (3) + E (4) = 11

• Calculation: $18 + 11 = \mathbf{29}$

• Result: The words "TWENTY NINE" are made of 29 straight lines, which equals its value.
  
  

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In the answer supplied to the puzzle, only 29 is listed and not 16 as well. Why I'm not sure but the letters of 29 are made up ONLY of straight line segments whereas 16 contains the curved letter S.